Simple instance of borel set and topology set

Question

To understand topology set and borel set and open set I need some concrete instance. Suppose $X=\left[0,1\right],A=\left[0,\frac{1}{2}\right]$, what is the smallest topology and Borel sigma algebra contains A in X?

Answer 1

This requires a bit clarification. First, note that given a set $X$, a topology $\tau$ on $X$ is a collection of subsets of $A$ satisfying the following three axioms:

• $\emptyset \in \tau$

• For any collection $\{A_{\alpha}\}$ in $\tau$, $\bigcup_{\alpha}A_{\alpha}\in \tau$

• For any finite collection $\{A_1,\dots,A_n\}\subset \tau$, $\bigcap_{i=1}^n A_i \in \tau$.

Note that, we 'define' members of topology to be open sets with respect to this particular topology.

A sigma algebra $\mathcal{F}$, is another collection of subsets of $X$, which satisfies

• $\emptyset, X \in \mathcal{F}$

• For every $X \in \mathcal{F}$, $X^c \in \mathcal{F}$

• For a countable collection $\{A_i\}_{i=1}^{\infty}\in\mathcal{F}$, $\bigcup_{i=1}^{\infty}A_i\in\mathcal{F}$.

Next, we will define Borel sigma algebra. Given any set $X$ and a topology $\tau$, the Borel sigma algebra is the smallest sigma algebra containing all members of the topology $\tau$. We call members of Borel $\sigma-$algebra as Borel sets.

Given this, the smallest topology containing $A$ is $\{\emptyset, X, A\}$; and the corresponding Borel sigma algebra is $\{\emptyset, X , A, A^c\}$.